Global Hopf bifurcation and permanence of a delayed SEIRS epidemic model

In this paper, an SEIRS system with two delays and the general nonlinear incidence rate is considered. The positivity and boundedness of solutions are investigated. The basic reproductive number, R 0 , is derived. If R 0 ? 1 , then the disease-free equilibrium is globally asymptotically stable and the disease dies out. If R 0 1 , then there exists a unique endemic equilibrium whose locally asymptotical stability and the existence of local Hopf bifurcations are established by analyzing the distribution of the characteristic values. An explicit algorithm for determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions is derived by using the center manifold and the normal form theory. Furthermore, there exists at least one positive periodic solution as the delay varies in some regions by using the global Hopf bifurcation result of Wu for functional differential equations. If R 0 1 , then the sufficient conditions of the permanence of the system are obtained, i.e., the disease eventually persists in the population. Especially, the upper and lower boundaries that each population can coexist are given exactly. Some numerical simulations are performed to confirm the correctness of theoretical analyses.

[1]  Junjie Wei,et al.  Bifurcation analysis of a population model and the resulting SIS epidemic model with delay , 2006 .

[2]  Wendi Wang,et al.  Global behavior of an SEIRS epidemic model with time delays , 2002, Appl. Math. Lett..

[3]  Jianhong Wu.  SYMMETRIC FUNCTIONAL DIFFERENTIAL EQUATIONS AND NEURAL NETWORKS WITH MEMORY , 1998 .

[4]  Rui Xu,et al.  Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity , 2010, Comput. Math. Appl..

[5]  Yukihiko Nakata,et al.  On the global stability of a delayed epidemic model with transport-related infection , 2011, Nonlinear Analysis: Real World Applications.

[6]  Yanni Xiao,et al.  Dynamical behavior for a stage-structured SIR infectious disease model , 2002 .

[7]  Yukihiko Nakata,et al.  Global stability of sirs epidemic models with a class of nonlinear incidence rates and distributed delays , 2012 .

[8]  Zhidong Teng,et al.  Impulsive Vaccination of an SEIRS Model with Time Delay and Varying Total Population Size , 2007, Bulletin of mathematical biology.

[9]  Y. N. Kyrychko,et al.  Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate , 2005 .

[10]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[11]  Mei Song,et al.  Global stability of an SIR epidemicmodel with time delay , 2004, Appl. Math. Lett..

[12]  Y. Kuang.  Delay Differential Equations: With Applications in Population Dynamics , 2012 .

[13]  James S. Muldowney,et al.  On Bendixson′s Criterion , 1993 .

[14]  Xiaofan Yang,et al.  Global stability of a delayed SIRS model with temporary immunity , 2008 .

[15]  Xiao-Qiang Zhao,et al.  Threshold Dynamics in a Delayed SIS Epidemic Model , 2001 .

[16]  Yang Kuang,et al.  Geometric Stability Switch Criteria in Delay Differential Systems with Delay Dependent Parameters , 2002, SIAM J. Math. Anal..

[17]  Yasuhiro Takeuchi,et al.  Global Stability for Delay SIR and SEIR Epidemic Models with Nonlinear Incidence Rate , 2010, Bulletin of mathematical biology.

[18]  Junjie Wei,et al.  Stability and bifurcation analysis in a delayed SIR model , 2008 .

[19]  Zhidong Teng,et al.  Global behavior and permanence of SIRS epidemic model with time delay , 2008 .

[20]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.